Whether a horizon \(H\)  is a timelike membrane, a dynamical horizon or an isolated horizon (or in general the causal character of an marginally trapped tube) can be determined by the function (see the references [1]}, [2]} for more details)
\(C=\frac{\mathcal {L}_k\Theta _k}{ \mathcal {L}_l\Theta _k},\) 
